Partial response channels which are well matched to a magnetic recording channel at low normalized densities are dominated by error-events involving a single channel symbol error, where the channel symbol represents the polarity of a write current +1 or -1. In partial response channels that are well matched to the magnetic recording channel at high transition densities, a different error-event is dominant. This dominant error-event involves three channel symbol errors which correspond to mistaking e.g. a channel symbol sequence of +1 -1 +1 for -1 +1 -1, or vice versa.
Coding is used to achieve improved performance of partial response channels. After characterizing a magnetic recording partial response channel, a list of dominant error-events is compiled. Then, a code is designed so that it does not contain any two sequences that are separated by a single error-event in the list. Merely constraining the sequences to a given coding constraint is not enough to obtain a significant coding gain. Rather, a Viterbi detector which is matched to the combination of the channel and the code must be used to insure that the detected sequence is allowed by the constraint.
Run-length limited codes that eliminate consecutive transitions have been suggested for high density magnetic recording. These codes are typically referred to as d=1 codes. By removing consecutive transitions, the +1 -1 +1 and -1 +1 -1 write current sequences are not allowed. Therefore, the dominant error-event at high densities is removed along with many other likely error-events. Because the code reduces the number of transitions and increases the spacing between transitions, magnetic write precompensation can be more accurately performed, leading to a reduction in non-linear transition shift. The most common of these codes is a rate 2/3 (d=1, k=7) code that has previously been used in magnetic recording channels employing peak detection techniques. More recently, a rate 4/5 code that removes the same error-events in an E.sup.2 PR4 magnetic recording channel has been proposed, R. Karabed and P. Siegel, "Coding for Higher Order Partial Response Channels", SPIE, Vol. 2605, 1996, pp. 115-126.
To achieve higher rates than are possible with a d=1 code, maximum transition ("MTR") codes that permit two, but not three, consecutive transitions have been suggested, and a plurality of MTR codes including a rate 16/19 (d=0, k=7) MTR block code have been proposed by J. Moon and B. Brickner in "Maximum Transition Run Codes for Data Storage Systems", 1996 Digests of Intermag '96, HB-10, April 1996, 3 pages. Their basic idea is to eliminate certain input bit patterns that would cause most error-events in a sequence detector. More specifically, the write current sequences in the proposed MTR code are not allowed to contain +1 -1 +1 or -1 +1 -1. There are only four possible ways of getting the dominant error-event at high linear densities. These four ways correspond to mistaking the following write current sequences, one for the other and conversely:
______________________________________ CASE 1. +1 +1 -1 +1 +1 2 consecutive transitions +1 -1 +1 -1 +1 4 consecutive transitions CASE 2. -1 +1 -1 +1 -1 4 consecutive transitions -1 -1 +1 -1 -1 2 consecutive transitions CASE 3. -1 +1 -1 +1 +1 3 consecutive transitions -1 -1 +1 -1 +1 3 consecutive transitions CASE 4. +1 +1 -1 +1 -1 3 consecutive transitions +1 -1 +1 -1 -1 3 consecutive transitions. ______________________________________
The idea of MTR coding is to eliminate three or more consecutive transitions, but allow the dibit pattern in the written magnetization waveform. Since at least one sequence in each of these four cases contains three or more consecutive transitions, the MTR code satisfies the condition that no two coded sequences are separated by the dominant high density error-event. With the MTR constraint, precompensation can be performed more accurately (i.e., mainly directed to dibit transitions), leading to a reduction in non-linear transition shift, but not as much of a reduction as can be obtained with a d=1 code that contains only isolated transitions.
The nomenclature used to represent the write current sequence is referred to as "non-return-to-zero" or "NRZ" notation. The number of states in the MTR constraint graph is cut in half by using "non-return-to-zero-inverse" or "NRZI" notation, where a zero corresponds to no transition, and a one corresponds to a transition. In NRZI notation the code constraint forbids the 111 sequence, which corresponds to the write current sequences +1 -1 +1 -1 and -1 +1 -1 +1. The MTR constraint graph is shown in FIG. 1 using NRZI notation.
Shannon's theorem states that it is possible, in principle, to devise a mechanism whereby a channel will transmit information with an arbitrarily small probability of error provided that the information rate is less than or equal to a rate called "channel capacity" or C. The Shannon channel capacity of a constraint graph is the logarithm base 2 of the largest real eigen value of the state transition matrix. The capacity of the FIG. 1 graph determines an upper bound on the maximum rate of a code satisfying that constraint. From the state transition matrix in Equation 1, the MTR constraint has a channel capacity C=0.8792. ##EQU1##
In general, the MTR constraint that allows at most two transitions in a row removes branches and/or states from the Viterbi detector for the channel. Therefore, an MTR code provides coding gain at high densities without adding complexity to the system. In a 16-state Viterbi detector, states +1 -1 +1 -1 and -1 +1 -1 +1 are simply removed, leaving only 14 states in the detector trellis. In an 8-state Viterbi detector states +1 -1 +1 and -1 +1 -1 each have only one branch entering and one branch leaving.
While MTR codes provide coding gain without adding complexity, a hitherto unsolved need has remained for a new type of modulation code for high density magnetic recording having a higher rate than the prior MTR block codes described by Moon and Brickner, for example, and without significantly increasing system complexity.